Concrete mathematics : a foundation for computer science

68 INTEGER FUNCTIONS
below the line f(x) = x:
We see from the graph that., for example,
lel = 2 ,
Tel = 3,
l-ej =-3,
r-e] = -2,
since e := 2.71828.. . .
By staring at this illustration we can observe several facts about floors
and ceilings. First, since the floor function lies on or below the diagonal line
f(x) = x, we have 1x1 6 x; similarly [xl 3 x. (This, of course, is quite
obvious from the definition.) The two functions are equal precisely at the
integer points:
lx] = x * x is an integer [xl = x.
(We use the notation ‘H’ to mean “if and only if!‘) Furthermore, when
they differ the ceiling is exactly 1 higher than the floor:
[xl - 1x1 = [x is not an integer] . (3.2) Cute.
By Iverson ‘s bracket
If we shift the diagonal line down one unit, it lies completely below the floor
function, so x - 1 < 1x1; similarly x + 1 > [xl. Combining these observations
gives us
conventions
complete
this is a
equation.
x-l < lx] 6 x 6 [xl < x+1. (3.3)
Finally, the functions are reflections of each other about both axes:
l-XJ = -[xl ; r-x.1 = -1xJ (3.4)